Editing Affine transformation (section)

== Image transformation ==
In their applications to [[digital image processing]], the affine transformations are analogous to printing on a sheet of rubber and stretching the sheet's edges parallel to the plane. This transform relocates pixels requiring intensity interpolation to approximate the value of moved pixels, bicubic [[interpolation]] is the standard for image transformations in image processing applications. Affine transformations scale, rotate, translate, mirror and shear images as shown in the following examples:<ref>{{cite book
  |last = Gonzalez
  |first = Rafael
  |title = 'Digital Image Processing, 3rd'
  |publisher = Pearson Hall
  |date = 2008
  |isbn = 9780131687288
}}</ref>

{| class="wikitable"
|-
! Transformation name
! Affine matrix
! Example
|-
| '''[[Identity operation|Identity]]''' (transform to original image)
| align="center" | <math>
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
</math>
| align="center" | [[File:Checkerboard identity.svg]] <!-- align="center", because the images depict the coordinate origin (concerning the matrices) in the center. -->
|-
| '''[[Translation (geometry)|Translation]]'''
| align="center" | <math>
\begin{bmatrix}
1 & 0 & v_x > 0\\
0 & 1 & v_y = 0\\
0 & 0 & 1
\end{bmatrix}
</math>
| align="right" | [[File:Checkerboard identity.svg]] <!-- There is no image for translation. align="right" works, because the image for scaling makes the column wide enough. -->
|-
| '''[[Reflection (mathematics)|Reflection]]'''
| align="center" | <math>
\begin{bmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\ 
0 & 0 & 1
\end{bmatrix}
</math>
| align="center" | [[File:Checkerboard reflection.svg]]
|-
| '''[[Scaling (geometry)|Scale]]'''
| align="center" | <math>
\begin{bmatrix}
c_x=2 & 0 & 0 \\
0 & c_y=1 & 0 \\ 
0 & 0 & 1
\end{bmatrix}
</math>
| [[File:Checkerboard scale.svg]]
|-
| '''[[Rotate]]'''
| align="center" |<math>
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) & 0 \\
\sin(\theta) & \cos(\theta) & 0 \\ 
0 & 0 & 1
\end{bmatrix}
</math>
| align="center" | [[File:Checkerboard rotate.svg]] <br/> where {{math|''θ'' {{=}} {{sfrac|π|6}} {{=}}30°}}
|-
| '''[[Shear matrix|Shear]]'''
| align="center" | <math>
\begin{bmatrix}
1 & c_x=0.5 & 0 \\
c_y=0 & 1 & 0 \\ 
0 & 0 & 1
\end{bmatrix}
</math>
| align="center" | [[File:Checkerboard shear.svg]]
|-
|}

The affine transforms are applicable to the registration process where two or more images are aligned (registered). An example of [[image registration]] is the generation of panoramic images that are the product of multiple images [[Image stitching|stitched]] together.

=== Affine warping  ===

The affine transform preserves parallel lines. However, the stretching and shearing transformations warp shapes, as the following example shows:

{| class="wikitable"
|-
| [[File:White_on_black_circle_image_256_by_256.png]]
| [[File:Affine_transform_sheared_circle.png]]
|}

This is an example of image warping. However, the affine transformations do not facilitate projection onto a curved surface or [[Distortion (optics)|radial distortions]].